Theorem REF  was proven for the Kuramoto model by Ha, Ha, and Kim [1]} and Choi, Ha, Jung, and Kim [2]}, with \(c_1=\frac{\pi }{4}\) . Namely, if the particles all lie on the same quarter-circle, then they stay within a quarter-circle for all time, their relative phases \(\theta _i-\theta _j\)  converge to a unique value, \(\theta _i-\theta _i^\infty -\nu t\)  converges to zero for some speed \(\nu \)  and some phase-locked state \(\lbrace \theta _i^\infty \rbrace \)  unique up to an additive constant, and \(\dot{\theta }_i-\dot{\theta }_j\)  converges exponentially to zero (one can extend the quarter-circle to a half-circle with some complications).
